Ch.13 Kinetics of a Particle - Force and Acceleration

13. Kinetics of a Particle – Force and Acceleration

HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Engineering Mechanics – Dynamics 13.01 Kinetics of a Particle: Force and Acceleration

Chapter Objectives• To state Newton’s Second Law of Motion and to define mass

and weight• To analyze the accelerated motion of a particle using the

equation of motion with different coordinate systems• To investigate central-force motion and apply it to problems in

space mechanics



§1.Newton’s Second Law of Motion- The branch of dynamics that deals with the relationship

between the change in motion of an object and the forces that cause this change

- The basis for kinetics is form Newton’s 2nd law (1686)

: unbalanced force acts on a particle: the mass of the particle: acceleration of the particle

- Mass is the quantity of matter an object has, it is a scalar. The mass provides the resistance of the object to any change in its velocity, that is its inertia



§1.Newton’s Second Law of Motion- The Newton’s 2nd law

• One of the most important equations in modern physics• A cornerstone of our modern built environment, and engineering• Validity based solely on experimental evidence

- The Newton’s second law – Disclaimer !• Albert Einstein (1905) showed that time is relative, as a result

Newton’s 2nd law fails to accurately describe the motion of objects traveling near the speed of light

• Advances in the Quantum physics have shown that motion of atoms and subatomic particles do not obey Newton’s 2nd law

• For most problems in modern engineering however neither of the above are applicable



§1.Newton’s Second Law of Motion- Newton’s Law of Gravitational Attraction

: force of attraction between the two particles, : universal constant of gravitation,

: mass of each of the two particles, : distance between the centers of the two particles,



§1.Newton’s Second Law of Motion- For an object “near” the earth surface

the force of attraction the weight () of the object

: the mass of the earth, : mass of the object, : gravitational acceleration,



§2.The Equation of Motion- When more than one force acts on an object, the resultant

force is vector summation of all the forces

- Consider an object of mass subject to the action of two forces and . If then , the object will either

• remain or rest, • have constant velocity,

Newton's 1st law of motion- Graphically, the free-body diagram the kinetic diagram



§2.The Equation of Motion- Inertial reference frame

• The acceleration of the particle should be measured with respect to a reference frame that is either fixed or translates with a constant velocity

• In this way, the observer will not accelerate and measurements of the particle’s acceleration will be the same from any reference of this type

Such a frame of reference is commonly known as a Newtonian or inertial reference frame



§3.Equation of Motion for a System of Particles- The equation of motion fro the particle

: resultant external force: resultant internal force

- If all these equations are added together vectorially

: summation of the internal forces: total mass of all the particles: center of mass acceleration

- The equation of motion



§4.Equations of Motion: Rectangular CoordinatesWhen a particle is moving relative to an

inertial ,, frame of reference, the forces acting on the particle, as well as its acceleration, may be expressed in terms of their , , components

- The equation of motion

- The scalar equations



§4.Equations of Motion: Rectangular Coordinates- Ex.13.1 The crate rests on a horizontal

plane for which the coefficient of kinetic friction is . If the crate is subjected to a towing force as shown, determine the velocity of the crate in starting from rest

SolutionFree body diagramEquation of motion

: (1): (2)

Solving (1) and (2)



§4.Equations of Motion: Rectangular CoordinatesEquation of motion

, Kinematicsthe applied force is constant the acceleration is constantthe initial velocity is zero the velocity of the crate in is



§4.Equations of Motion: Rectangular Coordinates- Ex.13.2 A projectile is fired vertically upward

from the ground with . Determine the maximum height to which it will travel if atmospheric resistance is (a) neglected; and (b) measured as , is the speed at any instant (Solution(a) Atmospheric resistance is neglected

Free body diagramEquation of motion

: Kinematics


⟹h=127𝑚HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

§4.Equations of Motion: Rectangular Coordinates(b) Atmospheric resistance is measured

Free body diagramEquation of motion

:

KinematicsThe acceleration is not constant since

:


⟹𝑧=114𝑚HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

§4.Equations of Motion: Rectangular Coordinates- Ex.13.3 The baggage truck

has a weight of and tows a cart and a cart . For a short time the driving frictional force developed at the

wheels of the truck is . If the truck starts from rest, determine its speed in . Also, what is the horizontal force acting on the coupling between the truck and cart at this instant? Neglect the size of the truck and carts

Solution• Free body diagram

Note: total weight Equation of motion

:



§4.Equations of Motion: Rectangular CoordinatesKinematics

The acceleration is a function of time:

• Free body diagramEquation of motion

:



§4.Equations of Motion: Rectangular Coordinates- Ex.13.4 A smooth collar is attached to a

spring having a stiffness and an unstretched length of . If the collar is released from rest at , determine its acceleration and the normal force of the rod on the collar at


: (1): (2)

(3)

(4)Giving the solution ,



§4.Equations of Motion: Rectangular Coordinates- Ex.13.6 The block is released from rest. If

the masses of the pulleys and the cord are neglected, determine the speed of the block in .


: (1): (2)

Kinematic

(3), ,

:



, ,

Fundamental Problems- F13.1 The motor winds in the cable with a constant

acceleration, such that the crate moves a distance in , starting from rest. Determine the tension developed in the cable. The coefficient of kinetic friction between the crate and the plane is



Fundamental Problems- F13.2 If motor exerts a force of on the cable, determine the

velocity of the crate when . The coefficients of static and kinetic friction between the crate and the plane are and , respectively. The crate is initially at rest

SolutionMaximum friction force

: the crate starts to move when applying :



Fundamental Problems- F13.3 A spring of stiffness is mounted against the block. If the

block is subjected to the force of , determine its velocity at . When , the block is at rest and the spring is uncompressed. The contact surface is smooth

Solution:



Fundamental Problems- F13.4 The car is being towed by a winch. If the winch exerts a

force of on the cable, where is the displacement of the car in meters, determine the speed of the car when , starting from rest. Neglect rolling resistance of the car



Fundamental Problems- F13.5 The spring has a stiffness and is unstretched when the

block is at . Determine the acceleration of the block when . The contact surface between the block and the plane is smooth

Solution

:



Fundamental Problems- F13.6 Block rests upon a smooth surface. If the coefficients of

static and kinetic friction between and are and respectively, determine the acceleration of each block if

SolutionConsider the motion of blocks and

:

Check if slipping occurs between block and block :



§5.Equations of Motion: Normal and Tangential Coordinates- The equation of motion for the particle may be written in the

tangential, normal, and binormal directions inertial

In scalar Recal



§5.Equations of Motion: Normal and Tangential Coordinates- Ex.13.6 Determine the banking

angle for the race track so that the wheels of the racing cars will not have to depend upon friction to prevent any car from sliding up or down the track. Assume the cars have negligible size, a mass , and travel around the curve of radius with a constant speed


: (1): (2)



§5.Equations of Motion: Normal and Tangential Coordinates- Ex.13.7 The disk is attached to the end of a cord. The

other end of the cord is attached to a ball-and-socket joint located at the center of a platform. If the platform rotates rapidly, and the disk is placed on it and released from rest as shown, determine the time it takes for the disk to reach a speed great enough to break the cord. The maximum tension the cord can sustain is , and the coefficient of kinetic friction between the disk and the platform is

Solution

Free body diagram



§5.Equations of Motion: Normal and Tangential CoordinatesEquation of motion

: (1): (2): (3)

Setting , solving the above equations, ,

KinematicsSince is constant, the time needed to break the cord

Thus



§5.Equations of Motion: Normal and Tangential Coordinates- Ex.13.8 Design of the ski jump shown

in the photo requires knowing the type of forces that will be exerted on the skier and her approximate trajectory. Determine the normal force on the skier the instant she arrives at the end of the jump, point , where her velocity is . What is her acceleration at this point

SolutionFree body diagram

slope at is horizontal

Equations of motion: (1): (2)


𝑑 𝑦𝑑𝑥= 1

100𝑥|

𝑥=0=0


§5.Equations of Motion: Normal and Tangential CoordinatesThe radius of curvature

Substituting this into Eq.(1) and solving for

Kinematics



§5.Equations of Motion: Normal and Tangential Coordinates- Ex.13.9 The skateboarder coasts

down the circular track. If he starts from rest when , determine the magnitude of the normal reaction the track exerts on him when . Neglect his size for the calculation

SolutionFree body diagramEquations of motion

: (1): (2)



§5.Equations of Motion: Normal and Tangential CoordinatesEquations of motion

: (1):

KinematicsSince , using to determine the speed of skateboarder

Substituting this result and into Eq.(1)



Fundamental Problems- F13.7 The block rests at a distance of from the center of the

platform. If the coefficient of static friction between the block and the platform is , determine the maximum speed which the block can attain before it begins to slip. Assume the angular motion of the disk is slowly increasing



Fundamental Problems- F13.8 Determine the maximum speed that the jeep can travel

over the crest of the hill and not lose contact with the road



Fundamental Problems- F13.9 A pilot weighs and is traveling at a constant speed of .

Determine the normal force he exerts on the seat of the plane when he is upside down at . The loop has a radius of curvature of

Solution:



Fundamental Problems- F13.10 The sports car is traveling along a banked road

having a radius of curvature of . If the coefficient of static friction between the tires and the road is , determine the maximum safe speed so no slipping occurs. Neglect the size of the car



Fundamental Problems- F13.11 If the ball has a velocity of when it is at the

position , along the vertical path, determine the tension in the cord and the increase in the speed of the ball at this position



Fundamental Problems- F13.12 The motorcycle has a mass of and a negligible

size. It passes point traveling with a speed of , which is increasing at a constant rate of . Determine the resultant frictional force exerted by the road on the tires at this instant



§6.Equations of Motion: Cylindrical Coordinates- When all the forces acting on a particle are

resolved into cylindrical components, i.e., along the unit vector directions , , , the equation of motion can be expressed as

In scalar



§6.Equations of Motion: Cylindrical Coordinates- Tangential and Normal forces

• Determination of the resultant force components, ,

causing a particle to move with a known acceleration• If acceleration is not specified at given instant, directions or

magnitudes of the forces acting on the particle must be known or computed to solve



§6.Equations of Motion: Cylindrical Coordinates• Consider the force that causes the particle to move along a

path

+The normal force which the path exerts on the particle is always perpendicular to the tangent of the path

+Frictional force always acts along the tangent in the opposite direction of motion



§6.Equations of Motion: Cylindrical Coordinates• The directions of and can be specified relative to the radial

coordinate by using the angle , which is defined between the extended radial line and the tangent to the curve

+If is positive, it is measured from the extended radial line to the tangent in a CCW sense or in the positive direction

+If it is negative, it is measured in the opposite direction to positive



§6.Equations of Motion: Cylindrical Coordinates- Ex.13.10 The smooth double-collar

can freely slide on arm and the circular guide rod. If the arm rotates with a constant angular velocity of , determine the force the arm exerts on the collar at the instant . Motion is in the horizontal plane

SolutionFree body diagramEquations of motion

: (1): (2)



acceleration: radial component , transverse component

§6.Equations of Motion: Cylindrical Coordinates

At The acceleration of the particle

Substituting these results into Eq.s (1)-(2) and solving to get,


⟹ { 𝑟=+0.5657𝑚�̇�=−1.6971𝑚 /𝑠�̈�=−5.091𝑚/ 𝑠2


§6.Equations of Motion: Cylindrical Coordinates- Ex.13.11 The smooth cylinder has a pin

through its center which passes through the slot in arm . If the arm is forced to rotate in the vertical plane at a constant rate , determine the force that the arm exerts on the peg at the instant

SolutionWhy is it a good idea to use polar coordinates to

solve this problem?Free body diagramEquations of motion

: (1): (2)



, , ,

§6.Equations of Motion: Cylindrical CoordinatesThe relation between an

At The acceleration of the particle

Solve Eq.s (1)-(2): ,


⟹ { 𝑟=+0.462𝑚�̇�=−0 .133𝑚/ 𝑠�̈�=+0 .192𝑚 /𝑠2


§6.Equations of Motion: Cylindrical Coordinates- Ex.13.12 A can , having a mass of ,

moves along a grooved horizontal slot. The slot is in the form of a spiral, which is defined by the equation , where is in radians. If the arm rotates with a constant rate in the horizontal plane, determine the force it

exerts on the can at the instant . Neglect friction and the size of the can

SolutionFree body diagram



§6.Equations of Motion: Cylindrical Coordinates

Equations of motion: (1): (2)



§6.Equations of Motion: Cylindrical CoordinatesKinematics

The time derivatives of and

At The acceleration of the can

Substituting these results into Eq.s (1)-(2) and solving to get

,


⟹ {𝑟=0.314𝑚�̇�=0 .4𝑚/ 𝑠�̈�=0𝑚 /𝑠2


Fundamental Problems- F13.13 Determine the constant angular velocity of the

vertical shaft of the amusement ride if . Neglect the mass of the cables and the size of the passengers

Solution

(1)

(2)

(3)


acceleration: radial component , transverse component HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien

Fundamental Problems- F13.14 The ball is blown through the smooth vertical circular

tube whose shape is defined by , where is in radians. If rad, where is in seconds, determine the magnitude of force exerted by the blower on the ball when

Solution




Fundamental Problems, , , ,

,



Fundamental Problems- F13.15 The car is traveling along the curved road

described by , where is in radians. If a camera is located at and it rotates with an angular velocity of and an angular acceleration of at the instant , determine the resultant friction force developed between the tires and the road at this instant

Solution




Fundamental Problems, , , ,

Equations of motion



Fundamental Problems- F13.16 The pin is constrained to move in the smooth

curved slot, which is defined by the lemniscate . Its motion is controlled by the rotation of the slotted arm , which has a constant clockwise angular velocity of . Determine the force arm exerts on the pin when . Motion is in the vertical plane

Solution



§7.Central-Force Motion and Space MechanicsRefer to the textbook



Ch.13 Kinetics of a Particle - Force and Acceleration

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